Auger-ionized free-carriers in a one-dimensional semiconductor are predicted to result in a strong band-gap renormalization. Isolated single-walled carbon nanotubes (SWCNT) under high-intensity laser irradiation exhibit strong nonlinear photoluminescence (PL) due to exciton-exciton annihilation (EEA). The presence of exciton disassociation during the rapid Auger-ionization caused by EEA would lead to a strong nonlinear absorption. By simultaneously measuring SWCNT-PL and optical absorption of isolated SWCNT clusters in the PL saturation regime, we give evidence that Auger-ionized excitons do not disassociate but remain bound.